An airplane flies 200 km due west from city A to city B and then 280 km in the direction of 30.5° north of west from city B to city C.

(a) In straight-line distance, how far is city C from city A?
in km?

(b) Relative to city A, in what direction is city C?
how many° north of west

(1) Add the vector going from A to B and the vector going from B to C. The length of the resultant vector will tell you the distance from A to C.

(2) The direction of the resultant vector of step 1 is what you are looking for. The angle (measured north from west) is arctan (Change in y)/(-Change in x)
= arctan (280 sin 30.5)/(200 + 280 cos 30.5) = arctan [142.1/(200 +241.2)]
17.8 degrees

To solve this problem, we can break it down into two components: the westward distance traveled and the northward distance traveled. Let's calculate each component step by step.

(a) Straight-line distance from city A to city C:
We first need to determine the westward (horizontal) distance. The airplane flies 200 km due west from city A to city B. This distance is fixed and does not change.

Next, we need to find the northward (vertical) distance. The airplane travels 280 km in the direction 30.5° north of west from city B to city C. To find the northward component, we can use trigonometry (specifically, sine). The northward distance can be found using the equation:
Northward distance = horizontal distance * sine(angle)

Given:
Horizontal distance (from city B to city C) = 280 km
Angle (30.5° north of west) = 30.5°

Northward distance = 280 km * sine(30.5°)

Now, we can calculate the straight-line distance from city A to city C using the Pythagorean theorem:
Distance = sqrt((westward distance)^2 + (northward distance)^2)

Substituting the values we obtained:
Straight-line distance = sqrt((200 km)^2 + (280 km * sine(30.5°))^2)
Evaluate the expression to obtain the final answer in km.

(b) Relative direction from city A to city C:
To determine the relative direction, we need to find the angle between the westward direction and the line connecting city A and city C. This angle, measured north of west, can be found using trigonometry (specifically, inverse tangent or arctan).

To find the angle, we can use the equation:
Angle = arctan(northward distance / westward distance)

Given:
Northward distance = 280 km * sine(30.5°)
Horizontal distance = 200 km

Angle = arctan( (280 km * sine(30.5°)) / 200 km)
Evaluate the expression to obtain the angle in degrees north of west.

Note: Make sure to convert the angle from radians to degrees if required.